Results 51 to 60 of about 1,313 (102)

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

, 2012
For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2}, \sum_{k=0}^{p-1}(3k+
Calkin N. J., Foata D., Franel J., Franel J., Guo V. J.W., Koepf W., MacMachon P. A., Riordan J., Strehl V., Sun Z.-W., Victor J.W. Guo, Zagier D.   +11 more
core   +1 more source

Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus

, 2013
In this paper, we derive some identities of Bernoulli, Euler, and Abel polynomials arising from umbral calculus.MSC:05A10, 05A19.
Dae San Kim, Taekyun Kim, Sang-Hun Lee, S. Rim   +3 more
semanticscholar   +1 more source

Consequences of a sextuple‐product identity

International Journal of Mathematics and Mathematical Sciences, Volume 10, Issue 3, Page 545-549, 1987., 1987
A sextuple‐product identity, which essentially results from squaring the classical Gauss‐Jacobi triple‐product identity, is used to derive two trigonometrical identities. Several special cases of these identities are then presented and discussed.
John A. Ewell
wiley   +1 more source

The log-convexity of the poly-Cauchy numbers

, 2016
In 2013, Komatsu introduced the poly-Cauchy numbers, which generalize Cauchy numbers. Several generalizations of poly-Cauchy numbers have been considered since then. One particular type of generalizations is that of multiparameter-poly-Cauchy numbers. In
Komatsu, Takao, Zhao, Feng-Zhen
core   +1 more source

Fully degenerate poly-Bernoulli numbers and polynomials

Open Mathematics, 2016
In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers.
Kim Taekyun, Kim Dae San, Seo Jong-Jin
doaj   +1 more source

The 26 Wilf-equivalence classes of length five quasi-consecutive patterns [PDF]

Discrete Mathematics & Theoretical Computer Science, 2018
We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$.
Evan Chen, Shyam Narayanan
doaj   +1 more source

A Pipe Dream Perspective on Totally Symmetric Self-Complementary Plane Partitions

Forum of Mathematics, Sigma
We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams.
Daoji Huang, Jessica Striker
doaj   +1 more source

Fourier series of functions involving higher-order ordered Bell polynomials

Open Mathematics, 2017
In 1859, Cayley introduced the ordered Bell numbers which have been used in many problems in number theory and enumerative combinatorics. The ordered Bell polynomials were defined as a natural companion to the ordered Bell numbers (also known as the ...
Kim Taekyun, Kim Dae San, Jang Gwan-Woo, Jang Lee Chae   +3 more
doaj   +1 more source

Some congruences involving binomial coefficients

, 2015
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant ...
Cao, Hui-Qin, Sun, Zhi-Wei
core   +1 more source

Recurrence formulae for Apostol-Bernoulli and Apostol-Euler polynomials

, 2012
In this paper, using generating functions and combinatorial techniques, we extend Agoh and Dilcher’s quadratic recurrence formula for Bernoulli numbers in (Agoh and Dilcher in J.
Yuan He, Chunping Wang
semanticscholar   +1 more source